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Granville recently asked how the Mahler measure behaves in the context of polynomial dynamics. For a polynomial 
$f(z)=z^d+\cdots \in {\mathbb C}[z],\ \deg (f)\ge 2,$ we show that the Mahler measure of the iterates 
$f^n$ grows geometrically fast with the degree 
$d^n,$ and find the exact base of that exponential growth. This base is expressed via an integral of 
$\log ^+|z|$ with respect to the invariant measure of the Julia set for the polynomial 
$f.$ Moreover, we give sharp estimates for such an integral when the Julia set is connected.
Let 
$p:\mathbb{C}\rightarrow \mathbb{C}$ be a polynomial. The Gauss–Lucas theorem states that its critical points, 
$p^{\prime }(z)=0$, are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows: suppose that 
$p$ has 
$n+m$ roots, where 
$n$ are inside the unit disk, then 
$$\begin{eqnarray}\max _{1\leq i\leq n}|a_{i}|\leq 1~\text{and}~m~\text{are outside}~\min _{n+1\leq i\leq n+m}|a_{i}|\geq d>1+\frac{2m}{n};\end{eqnarray}$$
$p^{\prime }$ has 
$n-1$ roots inside the unit disk and 
$m$ roots at distance at least 
$(dn-m)/(n+m)>1$ from the origin and the involved constants are sharp. We also discuss a pairing result: in the setting above, for 
$n$ sufficiently large, each of the 
$m$ roots has a critical point at distance 
${\sim}n^{-1}$.
Let F = GF(q). To any polynomial G ∈ F[x] there is associated a mapping Ĝ on the set IF of monic irreducible polynomials over F. We present a natural and effective theory of the dynamics of Ĝ for the case in which G is a monic q-linearized polynomial. The main outcome is the following theorem.
Assume that G is not of the form 
, where l ≥ 0 (in which event the dynamics is trivial). Then, for every integer n ≥ 1 and for every integer k ≥ 0, there exist infinitely many μ ∈ IF. having preperiod k and primitive period n with respect to Ĝ.
Previously, Morton, by somewhat different means, had studied the primitive periods of Ĝ when G = xq – ax, α a non-zero element of F. Our theorem extends and generalizes Morton's result. Moreover, it establishes a conjecture of Morton for the class of q-linearized polynomials.
